Malta's Prime Minister Joseph Muscat is under increasing pressure to resign amid protests over the latest twist in the case of murdered investigative journalist Daphne Caruana Galizia.

The escalating murder investigation has rocked the tiny Mediterranean island and reached the highest rungs of the country's politics, with two ministers and Muscat's chief of staff Keith Schembri stepping down from their posts this week.

What's the latest?

Thousands of demonstrators took to the streets of the capital Valletta after Muscat refused to give immunity to the main suspect in the 2017 murder, tycoon Yorgen Fenech, to disclose what he knows about the case.

AFP reported sources had told the agency Fenech had identified Schembri as the mastermind behind Caruana Galizia's 2017 car bomb killing. Schembri was arrested on Tuesday, but his release on Thursday sparked accusations of a cover-up.

Caruana Galizia's family accused the prime minister of protecting his long-time chief of staff, demanding he hand power over to a deputy who didn't have a conflict of interest.

"We share Malta's shock and anger at the release of Keith Schembri," the family said in a statement. "At least two witnesses and multiple pieces of physical evidence implicate Schembri in the assassination of our wife and mother."

They accused Muscat of playing "judge, jury, and executioner in an assassination investigation that so far implicates three of his closest colleagues."

Muscat to resign?

The Times of Malta reported on Friday that had Muscat privately told associates he planned to resign imminently. The prime minister's spokesperson would neither confirm nor deny the report.

Publicly, Muscat said he would remain in power, telling reporters he wanted "this case to be closed under my watch."

Muscat insisted on Friday he had recused himself from the decision on whether to grant immunity to Fenech. The prime minister later said he had reported Fenech to the police for attempted blackmail, claiming the mogul had threatened to implicate him in the affair if he was not given a pardon.

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Protesters in Valletta on Friday night chanted "Mafia" and "murderers."

"The killing of Daphne Caruana Galizia was a terrorist mafia act carried out by our state," activist Manuel Delia told AFP. He added that Schembri could have been kept in custody over corruption, money laundering and obstruction of justice.

What's the background?

Fenech, a tycoon whose business interests span the energy and tourism sectors, was arrested on his yacht last week after an alleged middleman in the murder, taxi driver Melvin Theuma, was offered a pardon to identify those involved.

That arrest was followed swiftly by the resignation of Schembri and tourism minister Konrad Mizzi, while economy minister Chris Cardona said he was "suspending himself."

Caruana Galizia, a popular journalist and blogger described as a "one-woman WikiLeaks", exposed cronyism and sleaze within the country's political and business elite.

She was blown up by a car bomb on October 16, 2017.

Before she was killed, she had alleged that Schembri and Mizzi had been involved in corruption. She had named Cardona in a separate case.

(With input from Reuters, AFP)